Matrix evaluations of noncommutative rational functions and Waring problems

• Matej Brešar ^[1] ; Jurij Volˇciˇc ^[2]
  1. [1] University of Ljubljana

     University of Ljubljana

     Eslovenia

     
  2. [2] University of Auckland

     University of Auckland

     Nueva Zelanda

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 5, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-025-01098-7
• Enlaces
  • Texto completo
• Resumen

  • Let r be a nonconstant noncommutative rational function in m variables over an algebraically closed field K of characteristic 0. We show that for n large enough, there exists an X ∈ Mn(K)m such that r(X) has n distinct and nonzero eigenvalues. This result is used to study the linear and multiplicative Waring problems for matrix algebras. Concerning the linear problem, we show that for n large enough, every matrix in sln(K) can be written as r(Y ) − r(Z) for some Y , Z ∈ Mn(K)m. We also discuss variations of this result for the case where r is a noncommutative polynomial. Concerning the multiplicative problem, we show, among other results, that if f and g are nonconstant polynomials, then, for n large enough, every nonscalar matrix in GLn(K) can be written as f (Y ) · g(Z) for some Y , Z ∈ Mn(K)m.

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